Applied Biomathematics for Nucleic Acid Chemistry and Protein Folding: Quantitative Simulations

In Chapter 1, we have DNA as a kind of nucleic acid consisting of two strands which are made up of two Watson-Crick base pairs: adenine-thymine (AT) and guanine-cytosine (GC). There are three components of the total energy. These are the inharmonic stacking interaction, hydrogen bond interaction and kinetic energy. Morse potential is used to mimic the hydrogen bond interaction between bases on the opposite strands for the overlapping π electrons, when two neighboring bases move out of the stack. The AT pair has 2 hydrogen bonds and the GC pair has 3 of them. The π electrons obey Bose - Einstein (BE) statistics, and the overlapping of them results in quantum fluctuation. It will be shown that this can be simplified into < Δy(t)Δy(t) >= 2DqΔt type fluctuation between the base pairs. Thus, a metropolis algorithm can be developed for the total potential energy by superposing two potential energy terms as well as including the quantum fluctuation in terms of random displacement of the π electrons. So, one can calculate the melting temperature of base pairs.

In Chapter 2, we demonstrate an analytical analysis of a previously published research for a percolation simulation. In that research the effect of mutations on adaptability was investigated in a bit-string model of invading species in a random environment. However, analytical analysis was missing which will be the topic here. The Hausdorff dimensions are calculated for the fractals and the conditions on invasion are analyzed analytically by manipulation of partial differential equations. Thus, various conclusions may be reached without having to run long simulations. 

 In Chapter 3, we investigate the folding dynamics of the plant-seed protein Crambin in a liquid environment, that usually happens to be water with some certain viscosity. To take into account the viscosity, necessitates a stochastic approach. This can be summarized by a 2D-Langevin equation, even though the simulation is still carried out in 3D. Solution of the Langevin equation will be the basic task in order to proceed with a Molecular Dynamics simulation, which will accompany a delicate Monte Carlo technique. The potential wells, used to engineer the energy space assuming the interaction of monomers constituting the protein-chain, are simply modeled by a combination of two parabola. This combination will approximate the real physical interactions, that are given by the well known Lennard-Jones potential. Contributions to the total potential from torsion, bending and distance dependent potentials are good to the fourth nearest neighbor. The final image is in very good geometric agreement with the real shape of the protein chain, which can be obtained from the protein data bank. The quantitative measure of this agreement is the similarity parameter with the native structure, which is found to be 0.91 < 1 for the best sample. The folding time can be determined from Debye-relaxation process. We apply two regimes and calculate the folding time, corresponding to the elastic domain mode, which yields 5.2ps for the same sample. 

In Chapter 4, we have studied the chain length dependence of folding time for proteins by implementing a novel Monte Carlo (MC) method. The physical parameters in our model are derived from the statistics for bending and torsion angles and distances between the centers of the monomers up to the fourth neighborhood. By assigning potential wells to each of the physical parameters, we are able to use a modified Metropolis algorithm to efficiently trace the later conformations of the proteins as time evolves. Our prescription for microscopic dynamics for the protein "Crambin" results in an increase in folding times with increasing chain length. The folding times are determined via Debye relaxation process. 

In Chapter 5, DNA is a kind of nucleic acid consisting of two strands which are made up of two Watson-Crick base pairs: adenine-thymine (AT) and guanine-cytosine (GC). Vitrification (from Latin vitreum, "glass") on the other hand is the transformation of a substance into a glass. DNA vitrification is achieved by rapidly cooling DNA in a liquid state through the glass transition. The quantum fluctuation in terms of random displacement and specific heat capacity of the π electrons in hydrogen bonds was studied earlier to calculate the DNA melting temperature. Same principles along with the inclusion of longitudinal phonon vibrations will be used here in order to calculate the vitrification temperature (glass transition temperature) of base pairs. This has an important application in cryonics and cryopreservation.

In Appendix, we have an analytical investigation of the well-known 10-12 potential of hydrogen-hydrogen covalent bond. In this research, we will make an elaboration of the well-known 6-12 Lennard-Jones potential in case of this type of bond. Though the results are illustrated in many text books and literature, an analytical analysis of these potentials is missing almost everywhere. The power laws are valid for small radial distances, which are calculated to some extent. The internuclear separation as well as the binding energy of the hydrogen molecule are evaluated with success.